[Math] Value of sine of complex numbers

Question

I stumbled upon a problem with evaluating the sine function for complex arguments.
I know that in general I can use
$$
\sin(ix)=\frac{1}{2i}(\exp(-x)-\exp(x))=i\sinh(x).
$$
But I could also write the sine function as the imaginary part of the exponential function as
$$\sin(ix)=\text{Im}(\exp(i(ix)))=\text{Im}(\exp(-x))=0$$
where Im is the imaginary part.
Well, apparently I am not allowed to write it like that, but I don't see why. Could you give me a hint what went wrong here?

Answer 1

This, of course, uses three interconnected formulas: $e^{ix}= cos(x)+ i sin(x)$, $cos(x)= \frac{e^{ix}+ e^{-ix}}{2}$, and $sin(x)= \frac{e^{ix}- e^{-ix}}{2}$

Your error is that you are assuming that the imaginary part of $e^{ix}$ is "i sin(x)". That is true only if itself is real. If x is not real the $i sin(x)$ is not imaginary because sin(x) is not real.